which is the standard arc length of the trajectory \(x(t)\) . They prove the proportional relationship between these two complexity measures.
To obtain the estimated value of effective complexity, Raghu et al. [1]} bound the expected value of trajectory length of any layer in a deep ReLU neural network.
Specifically, given a deep ReLU neural network whose weights are initialized by \(N(0, \sigma ^2_w/m)\)  and the biases are initialized by \(N(0, \sigma _b^2)\) . Let \(z^{(i)}(x(t))\)  denote the new trajectory obtained after the transformation of the first \(i\)  hidden layers of the trajectory \(x(t)\) . The expected value of its trajectory length can be bounded by
\(\nonumber \mathbb {E}[l(z^{(i)}(x(t)))] \ge O(\frac{\sigma _w \sqrt{m}}{\sqrt{m+1}})^i l(x(t))\) 
